Two of England's Group A matches and all three of Scotland's, including the Wembley showdown between the sides on Saturday 15 June, are now officially sold out.
Ticket sales are also suspended for the match between Italy and Germany at Old Trafford on Wednesday 19 June and all three Group D matches at Nottingham Forest's City Ground.
The FA says that some tickets have been held back for competing nations, including up to an extra 2,000 for Scottish fans who want to be at Wembley to watch their team playing the "auld enemy" for the first time since 1989.
The Scots have so far only been guaranteed the regulation 7,000 seats that go to all the visiting countries for each of their matches.
An FA spokeswoman, Philippa Warhurst, said: "There is a possibility of more tickets being made available to Scotland fans. We are looking at security arrangements and we will see what we can do."
The FA has come in for heavy criticism from fans who have found the Euro 96 telephone lines jammed since Sunday's draw.
However, Warhurst said: "We have had 50 operators on the phones at all times and people have known that tickets have been on sale since January 1994. Hopefully now that we have announced that ticket sales have been frozen for certain games, it will unfreeze the lines for people wanting tickets for other matches."
EUROPEAN CHAMPIONSHIP Finals: Ticket sales suspended for the following matches: 10 June: Netherlands v Scotland (Villa Park); 11 June: Turkey v Croatia (City Ground); 14 June: Portugal v Turkey (City Ground); 15 June: Scotland v England (Wembley); 18 June: Scotland v Switzerland (Villa Park); Netherlands v England (Wembley); 19 June: Croatia v Portugal (City Ground); Turkey v Denmark (Hillsborough); Italy v Germany (Old Trafford).